As Donald J. Trump and his wife, Melania, arrived at the White House on Friday morning to begin inauguration ceremonies, Mrs. Trump presented the departing first lady, Michelle Obama, with a blue box from Tiffany & Company — a company that happens to have a flagship store next to the Trump Tower in Manhattan.

The gift’s arrival seemed unexpected, and it briefly created an awkward moment, as Mrs. Obama tried to figure out what to do with it. She turned this way and that way. The motionless Marine behind her didn’t look particularly interested in holding it. There was no table to put it on. What’s a departing first lady to do?

President Obama made an executive decision: He took the box, walked inside and handed it to an unknown person.

The moment was widely received on social media as a graceful parting gift from Mrs. Trump.